{"id":849,"date":"2023-11-10T23:19:48","date_gmt":"2023-11-10T15:19:48","guid":{"rendered":"http:\/\/ggapa.net:81\/?p=849"},"modified":"2024-02-12T14:40:12","modified_gmt":"2024-02-12T06:40:12","slug":"%e6%95%b0%e6%8d%ae%e7%bb%93%e6%9e%84%e5%ad%a6%e4%b9%a0%e7%ac%94%e8%ae%b0","status":"publish","type":"post","link":"http:\/\/ggapa.net:81\/2023\/11\/10\/%e6%95%b0%e6%8d%ae%e7%bb%93%e6%9e%84%e5%ad%a6%e4%b9%a0%e7%ac%94%e8%ae%b0\/","title":{"rendered":"\u6570\u636e\u7ed3\u6784\u5b66\u4e60\u7b14\u8bb0"},"content":{"rendered":"<h1>1 \u7ebf\u6bb5\u6811<\/h1>\n<h2>1.1 \u4ec0\u4e48\u662f\u7ebf\u6bb5\u6811\uff1f<\/h2>\n<p>\u7ebf\u6bb5\u6811\u5728\u7b97\u6cd5\u7ade\u8d5b\u4e2d\u5e38\u7528\u4e8e\u533a\u95f4\u64cd\u4f5c\u3002\u65f6\u95f4\u590d\u6742\u5ea6$O(\\log n)$\u3002<\/p>\n<h2>1.2 \u7ebf\u6bb5\u6811\u7684\u5e94\u7528<\/h2>\n<h3>1.2.1 \u7ebf\u6bb5\u6811\u7684\u57fa\u672c\u5e94\u7528<\/h3>\n<p>\u987e\u540d\u601d\u4e49\uff0c\u533a\u95f4\u4fee\u6539\uff0c\u533a\u95f4\u67e5\u8be2\u3002<\/p>\n<h3>1.2.2 \u7ebf\u6bb5\u6811\u6700\u5927\u5b57\u6bb5\u548c\u95ee\u9898\u4e0a\u7684\u8fd0\u7528<\/h3>\n<p>\u95ee\u9898\u63cf\u8ff0\uff1a<\/p>\n<ul>\n<li>\n<p>\u64cd\u4f5c\u4e00\uff1a\u8fdb\u884c\u5355\u70b9\u4fee\u6539\u3002<\/p>\n<\/li>\n<li>\n<p>\u64cd\u4f5c\u4e8c\uff1a\u7ed9\u5b9a $l$ \u548c $r$\uff0c\u95ee $[l, r]$ \u4e2d\u7684\u6700\u5927\u7684\u5b50\u6bb5\u548c\u662f\u591a\u5c11\uff1f<\/p>\n<\/li>\n<\/ul>\n<p>\u4e3a\u4e86\u89e3\u51b3\u8fd9\u4e2a\u95ee\u9898\uff0c\u6211\u4eec\u9700\u8981\u5f15\u5165\u7684\u61d2\u6807\u8bb0\u6709\u5982\u4e0b\u51e0\u4e2a<\/p>\n<ul>\n<li>$maxv$ \u533a\u95f4\u6700\u5927\u5b50\u6bb5\u548c\uff0c\u4e5f\u5c31\u662f\u95ee\u9898\u7684\u7b54\u6848\u3002<\/li>\n<li>$sumv$ \u533a\u95f4\u7684\u548c\u3002<\/li>\n<li>$maxnl$ \u5305\u542b\u5de6\u7aef\u70b9\u7684\u533a\u95f4\u6700\u5927\u5b50\u6bb5\u548c\u3002<\/li>\n<li>$maxr$ \u5305\u542b\u53f3\u7aef\u70b9\u7684\u533a\u95f4\u6700\u5927\u5b50\u6bb5\u548c\u3002<\/li>\n<\/ul>\n<p><strong>\u5728\u4fee\u6539\u65f6<\/strong><\/p>\n<p>\u5148\u66f4\u65b0 $maxv$\u3002<\/p>\n<p>\u82e5 $maxl &lt; 0 \\and maxr &lt; 0$\uff08\u60c5\u51b5\u4e00\uff09\uff0c$maxv = \\max{maxl,maxr}$ \u3002\u9898\u76ee\u4e2d\u8981\u6c42\u5fc5\u987b\u8981\u53d6\u4e00\u4e2a\u5143\u7d20\uff0c\u4e0d\u80fd\u4e0d\u53d6\u3002<\/p>\n<p>\u82e5\u60c5\u51b5\u4e00\u4e0d\u6210\u7acb\uff0c\u5219\u8bf4\u660e $maxl &lt; 0 \\or maxr &lt; 0$ \u4e3a\u771f\uff08\u60c5\u51b5\u4e8c\uff09\uff0c\u6b64\u65f6\u9700\u8981\u5c06 $maxv$ \u5f52 $0\uff0c\u7136\u540e\u5c06$ $maxv$ \u5c31\u53ef\u4ee5\u52a0\u4e0a $maxl,maxr$ \u4e2d\u4e3a\u6b63\u7684\u503c\u3002<\/p>\n<p>\u4e5f\u6709\u53ef\u80fd\u4ee5\u4e0a\u7684\u60c5\u51b5\u90fd\u4e0d\u662f\u6700\u5927\u5b50\u6bb5\u548c\uff0c\u6700\u540e\u7684 $maxv = \\max{maxv,maxv(\u5de6\u513f\u5b50),maxv(\u53f3\u513f\u5b50)}$\u3002\u7279\u522b\u8bf4\u660e\uff0c\u4ee5\u4e0a\u4e24\u79cd\u60c5\u51b5\u4e2d\u7684 $maxl$ \u548c $maxr$ \u5206\u522b\u662f\u5176\u53f3\u5de6\u513f\u5b50\u7684\u4fe1\u606f\u3002<\/p>\n<p>\u63a5\u7740\u66f4\u65b0 $maxr$ \u548c  $maxl$ \u3002<\/p>\n<p>\u56e0\u4e3a\u7ebf\u6bb5\u6811\u9700\u8981\u5408\u5e76\u4e24\u4e2a\u533a\u95f4\uff0c\u53ef\u4ee5\u5f97\u51fa $maxl$ \u4e3a\u5de6\u513f\u5b50\u7684 $maxl$ \u4e0e\u5de6\u513f\u5b50 $sumv$ \u548c\u53f3\u513f\u5b50 $maxl$ \u548c\u7684\u6700\u5927\u503c\u3002\u800c $maxr$ \u4e3a\u53f3\u513f\u5b50\u7684 $maxr$ \u4e0e\u53f3\u513f\u5b50 $sumv$ \u548c\u5de6\u513f\u5b50 $maxr$ \u7684\u6700\u5927\u503c\u3002\u8be5\u8282\u70b9\u7684 $sum$ \u4e3a\u5176\u5de6\u53f3\u513f\u5b50 $sum$ \u7684\u548c\u3002<\/p>\n<p><strong>\u5728\u67e5\u8be2\u65f6<\/strong><\/p>\n<p>\u82e5\u67e5\u8be2\u533a\u95f4\u8986\u76d6\u8fd9\u4e00\u8282\u70b9\uff0c\u5219\u8fd4\u56de\u8fd9\u4e00\u8282\u70b9\u7684\u4fe1\u606f\uff1b\u82e5\u5b8c\u5168\u8986\u76d6\u4e86\u5176\u4e2d\u4e00\u4e2a\u513f\u5b50\uff0c\u5219\u8fd4\u56de\u8986\u76d6\u5230\u7684\u513f\u5b50\u7684\u4fe1\u606f\uff1b\u82e5\u6ca1\u6709\u5b8c\u5168\u8986\u76d6\u4e00\u4e2a\u513f\u5b50\uff0c\u901a\u8fc7\u5408\u5e76\u7684\u65b9\u5f0f\u67e5\u8be2\u8986\u76d6\u5230\u7684\u533a\u95f4\u3002\u6700\u540e\u8fd4\u56de\u7684 $maxv$ \u5373\u4e3a\u4e00\u6b21\u67e5\u8be2\u7684\u6700\u7ec8\u7b54\u6848\u3002<\/p>\n<p><a href=\"https:\/\/www.luogu.com.cn\/record\/134060982\">\u6a21\u677f\u9898\u4ee3\u7801<\/a><\/p>\n<h3>1.2.3  01 \u533a\u95f4\u7ffb\u8f6c + \u6700\u957f\u5b50\u6bb5\u548c<\/h3>\n<p><a href=\"https:\/\/vjudge.net\/solution\/48986676\/tLDGx7Qe1w35AuTCfe6P\">Code<\/a><\/p>\n<h2>1.3 \u5229\u7528\u6811\u72b6\u6570\u7ec4\u4ee3\u66ff\u7ebf\u6bb5\u6811<\/h2>\n<p>\u4f17\u6240\u5468\u77e5\u7ebf\u6bb5\u6811\u7684\u5e38\u6570\u5927\uff0c\u4ee3\u7801\u96be\u5199\uff0c\u800c\u5728\u8003\u8bd5\u4e2d\u7ecf\u5e38\u51fa\u73b0\u65f6\u95f4\u4e0d\u8db3\u4ece\u800c\u5199\u4e0d\u51fa\u7ebf\u6bb5\u6811\u9020\u6210\u60b2\u5267\u7684\u60c5\u51b5\u3002\u4f46\u73b0\u5728\uff0c\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u6811\u72b6\u6570\u7ec4\u8fd9\u79cd\u4ee3\u7801\u597d\u5199\u7684\u6570\u636e\u7ed3\u6784\u4ee3\uff0c\u5b8c\u6210\u7c7b\u4f3c\u4e8e\u7ebf\u6bb5\u6811\u7684\u533a\u95f4\u4fee\u6539\u533a\u95f4\u67e5\u8be2\u529f\u80fd\u3002<\/p>\n<p>\u4e3a\u4e86\u8d4b\u4e88\u6811\u72b6\u6570\u7ec4\u8fd9\u4e00\u4f1f\u5927\u7684\u4f7f\u547d\uff0c\u6211\u4eec\u9700\u8981\u4e24\u4e2a\u6811\u72b6\u6570\u7ec4\uff0c$t1\u3001t2$\u3002<\/p>\n<p>\u4ee4 $t[ i ] = a[i] - a[i-1]$ (\u5dee\u5206\uff0c\u5176\u4e2d $a$ \u4e3a\u7ed9\u5b9a\u7684\u6570\u7ec4)\uff0c\u6613\u5f97\uff1a<\/p>\n<p>$$\\sum_{i=1}^{n} t1[i]=a[n]$$<\/p>\n<p>\u4e8e\u662f\u6211\u4eec\u53ef\u4ee5\u8fdb\u884c\u8f6c\u6362\uff1a<\/p>\n<p>$$=\\sum_{i=1}^{n}a[i] = \\sum_{i = 1}^{n}\\sum_{j=1}^{i}t1[j]$$<\/p>\n<p>\u7531\u52a0\u6cd5\u4ea4\u6362\u5f8b\u548c\u52a0\u6cd5\u7ed3\u5408\u5f8b\u53ef\u5f97\uff1a<\/p>\n<p>$$=\\sum_{i =1}^{n} t1[i]\\times(n-i+1)$$<\/p>\n<p>\u7531\u4e58\u6cd5\u5206\u914d\u5f8b\u53ef\u5f97\uff1a<\/p>\n<p>$$=n\\times \\sum_{i=1}^{n} t1[i]- \\sum_{i = 1}^{n}t1[i]\\times (i - 1)$$<\/p>\n<p>\u4ee4 $t2[i] = t1[i]\\times(i-1)$ \uff0c\u6700\u7ec8\u53ef\u4ee5\u5f97\u5230\uff1a<\/p>\n<p>$$=n\\times \\sum_{i=1}^{n} t1[i]- \\sum_{i = 1}^{n}t2[i]$$<\/p>\n<p>\u5bf9\u4e8e\u4e00\u6b21\u6dfb\u52a0\u64cd\u4f5c\uff1a<\/p>\n<pre><code class=\"language-cpp\">void modif(int x, int v) {\n    t2.modify(x, (x - 1) * v); t1.modify(x, v);\n}\n<\/code><\/pre>\n<p>\u5bf9\u4e8e\u4e00\u6b21\u67e5\u8be2\u64cd\u4f5c\uff1a<\/p>\n<pre><code class=\"language-cpp\">int qure(int y) {\n    return t1.qurey(y) * (y) - t2.qurey(y);\n}\n<\/code><\/pre>\n<p>\u90a3\u4e48\u5bf9\u4e8e\u4e00\u6b21\u533a\u95f4\u67e5\u8be2\uff0c\u5219\u6709\uff1a<\/p>\n<pre><code class=\"language-cpp\">int qurey(int l, int r) {\n    return qure(r) - qure(l  - 1);\n}<\/code><\/pre>\n<p>\u540c\u7406\uff0c\u533a\u95f4\u4fee\u6539\uff1a<\/p>\n<pre><code class=\"language-cpp\">void modify(int l, int r, int v) {\n    modif(l ,v); modif(r + 1, -v);\n}<\/code><\/pre>\n<p><a href=\"https:\/\/www.luogu.com.cn\/record\/144854741\">\u5b8c\u6574\u4ee3\u7801<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"<p>1 \u7ebf\u6bb5\u6811 1.1 \u4ec0\u4e48\u662f\u7ebf\u6bb5\u6811\uff1f \u7ebf\u6bb5\u6811\u5728\u7b97\u6cd5\u7ade\u8d5b\u4e2d\u5e38\u7528\u4e8e\u533a\u95f4\u64cd\u4f5c\u3002\u65f6\u95f4\u590d\u6742\u5ea6$O(\\log n)$\u3002 1. [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[12],"tags":[],"_links":{"self":[{"href":"http:\/\/ggapa.net:81\/wp-json\/wp\/v2\/posts\/849"}],"collection":[{"href":"http:\/\/ggapa.net:81\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/ggapa.net:81\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/ggapa.net:81\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/ggapa.net:81\/wp-json\/wp\/v2\/comments?post=849"}],"version-history":[{"count":8,"href":"http:\/\/ggapa.net:81\/wp-json\/wp\/v2\/posts\/849\/revisions"}],"predecessor-version":[{"id":1096,"href":"http:\/\/ggapa.net:81\/wp-json\/wp\/v2\/posts\/849\/revisions\/1096"}],"wp:attachment":[{"href":"http:\/\/ggapa.net:81\/wp-json\/wp\/v2\/media?parent=849"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/ggapa.net:81\/wp-json\/wp\/v2\/categories?post=849"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/ggapa.net:81\/wp-json\/wp\/v2\/tags?post=849"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}